Symplectic symmetric pairs of contragredient Lie superalgebras

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Let L be a contragredient Lie superalgebra. A symmetric pair of L is a pair $$({\mathfrak {g}},{\mathfrak {g}}^\sigma )$$ , where $${\mathfrak {g}}$$ is a real form of L, and $$\sigma $$ is a $${\mathfrak {g}}$$ -involution with invariant subalgebra $${\mathfrak {g}}^\sigma $$ . We show that a symmetric pair carries invariant symplectic forms if and only if $${\mathfrak {g}}^\sigma $$ has a 1-dimensional center. Furthermore, the symplectic form is pseudo-Kahler if and only if the center of $${\mathfrak {g}}^\sigma $$ is compact. As an application, we classify the symplectic symmetric pairs, as well as the pseudo-Kahler symmetric pairs.